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December  2016, 21(10): 3637-3654. doi: 10.3934/dcdsb.2016114

## Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays

 1 Vietnam Education Publishing House, 81 Tran Hung Dao, Hanoi, Vietnam 2 Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam 3 Department of Mathematics, Electric Power University, 235 Hoang Quoc Viet, Hanoi, Vietnam

Received  November 2015 Revised  August 2016 Published  November 2016

We deal with the Cauchy problem associated with integro-differential inclusions of diffusion-wave type involving infinite delays. Based on the behavior of resolvent operator associated with the linear part, an explicit estimate for solutions will be established. As a consequence, the weak stability of zero solution is proved in case the resolvent operator is asymptotically stable.
Citation: Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114
##### References:
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Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [28] T. I. Seidman, Invariance of the reachable set under nonlinear perturbations,, SIAM J. Control Optim., 25 (1987), 1173.  doi: 10.1137/0325064.  Google Scholar

show all references

##### References:
 [1] B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semi-linear Cauchy problems with non-dense domain,, Nonlinear Anal., 72 (2010), 3190.  doi: 10.1016/j.na.2009.12.016.  Google Scholar [2] N. T. Anh and T. D. Ke, Decay integral solutions for neutral fractional differential equations with infinite delays,, Math. Methods Appl. Sci., 38 (2015), 1601.  doi: 10.1002/mma.3172.  Google Scholar [3] E. Bazhlekova, Fractional Evolution Equations in Banach Spaces,, Ph.D. Thesis, (2001).   Google Scholar [4] T. A. Burton, Stability by Fixed Point Theory for Functional Differential Equations,, Dover Publications, (2006).   Google Scholar [5] T. A. Burton and T. Furumochi, Fixed points and problems in stability theory for ordinary and functional differential equations,, Dyn. Syst. Appl., 10 (2001), 89.   Google Scholar [6] J. P. Carvalho dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations,, Appl. Math. Lett., 23 (2010), 960.  doi: 10.1016/j.aml.2010.04.016.  Google Scholar [7] N. M. Chuong, T. D. Ke and N. N. Quan, Stability for a class of fractional partial integro-differential equations,, J. Integral Equations Appl., 26 (2014), 145.  doi: 10.1216/JIE-2014-26-2-145.  Google Scholar [8] C. Cuevas and J. César de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay,, Nonlinear Anal., 72 (2010), 1683.  doi: 10.1016/j.na.2009.09.007.  Google Scholar [9] C. Cuevas and J. César de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations,, Appl. Math. Lett., 22 (2009), 865.  doi: 10.1016/j.aml.2008.07.013.  Google Scholar [10] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,, Discrete Contin. Dyn. Syst., (2007), 277.   Google Scholar [11] J. Diestel, W. M. Ruess and W. Schachermayer, Weak compactness in $L^{1}(\mu, X)$,, Proc. Amer. Math. Soc., 118 (1993), 447.  doi: 10.2307/2160321.  Google Scholar [12] R. D. Driver, Ordinary and Delay Differential Equations,, Springer-Verlag, (1977).   Google Scholar [13] I. Ekeland and R. Temam, Convex Analysis and Variational Problems,, Society for Industrial and Applied Mathematics (SIAM), (1999).  doi: 10.1137/1.9781611971088.  Google Scholar [14] A. F. Filippov, Differential Equations with Discontinuous Righthand Sides,, Translated from the Russian. Mathematics and its Applications (Soviet Series), (1988).  doi: 10.1007/978-94-015-7793-9.  Google Scholar [15] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation,, Osaka J. Math., 27 (1990), 309.   Google Scholar [16] Y. Fujita, Integrodifferential equation which interpolates the heat equation and the wave equation. II,, Osaka J. Math., 27 (1990), 797.   Google Scholar [17] C. Gori, V. Obukhovskii, M. Ragni and P. Rubbioni, Existence and continuous dependence results for semilinear functional differential inclusions with infinite delay,, Nonlinear Anal., 51 (2002), 765.  doi: 10.1016/S0362-546X(01)00861-6.  Google Scholar [18] J. Hale and J. Kato, Phase space for retarded equations with infinite delay,, Funkcial. Ekvac., 21 (1978), 11.   Google Scholar [19] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993).  doi: 10.1007/978-1-4612-4342-7.  Google Scholar [20] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay,, Lecture Notes in Mathematics, (1473).  doi: 10.1007/BFb0084432.  Google Scholar [21] M. Kamenskii, V. Obukhovskii and P. Zecca, Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces,, in: de Gruyter Series in Nonlinear Analysis and Applications, (2001).  doi: 10.1515/9783110870893.  Google Scholar [22] F. Mainardi and P. Paradisi, Fractional diffusive waves,, J. Comput. Acoustics, 9 (2001), 1417.  doi: 10.1142/S0218396X01000826.  Google Scholar [23] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach,, Phys. Rep., 339 (2000), 1.  doi: 10.1016/S0370-1573(00)00070-3.  Google Scholar [24] R. Metzler and J. Klafter, Accelerating Brownian motion: A fractional dynamics approach to fast diffusion,, Europhys. Lett., 51 (2000), 492.  doi: 10.1209/epl/i2000-00364-5.  Google Scholar [25] J.-S. Pang and D. E. Stewart, Differential variational inequalities,, Math. Program. Ser. A, 113 (2008), 345.  doi: 10.1007/s10107-006-0052-x.  Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [27] K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems,, J. Math. Anal. Appl., 382 (2011), 426.  doi: 10.1016/j.jmaa.2011.04.058.  Google Scholar [28] T. I. Seidman, Invariance of the reachable set under nonlinear perturbations,, SIAM J. Control Optim., 25 (1987), 1173.  doi: 10.1137/0325064.  Google Scholar
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